Let $\displaystyle f$ be an analytic function on a disc D.

Is it true that:

$\displaystyle f$ is constant $\displaystyle \iff$ $\displaystyle f'(z) =0 $ $\displaystyle \forall z \in D $

with ' denoting the derivative wrt to complex variable z. Or do we have to include some conditions on the partial derivatives wrt x,y (where Re(z)=x and Im(z)=y) ?

Thanks for anyhelp!